Photorefractive Solitons (Chapter in Springer book ... - Tripod

Photorefractive Solitons 

E. DelRe 1 , M. Segev 2 , D. Christodoulides 3 , B. Crosignani 1 , and G. Salamo 5 

1 

Dipartimento di Fisica, Università dell’Aquila, 67010 L’Aquila, Italy and 

Istituto Nazionale Fisica della Materia - UdR Roma ”La Sapienza”, 00185 

Roma, Italy eugenio.delre@aquila.infn.it 

2 

Physics Department, Technion, Haifa, Israel msegev@tx.technion.ac.il 

3 

School of Optics-CREOL, University of Central Florida, Orlando, 32816-2700 

United States demetri@creol.ucf.edu 

4 

Physics Department, University of Arkansas, Fayetteville, 72703-0000 United 

States salamo@uark.edu 

1 Introduction 

Solitons are a universal phenomenon that appears in a wealth of systems in 

nature. The past few decades have witnessed their identification and observation 

in the more diverse physical systems: in shallow and deep water waves, 

charge density waves in plasma, sound waves in liquid helium, matter waves 

in Bose-Einstein condensates, excitations on DNA chains, ”branes” at the 

end of open strings in superstring theory, domain walls in supergravity, and 

many more. Solitons can even appear for electromagnetic waves in vacuum, 

where they are supported by QED nonlinearities. And, of course, in Optics: 

here solitons truly manifest themselves in a large variety of settings (for a review 

on optical solitons see Refs.[1, 2, 3, 4]). In all these diverse systems, that 

vary in almost everything from size to dimensionality, from underlying forces 

to physical mechanisms, propagation leads to a family of nonlinear waves - 

solitons - that have the same, universal, features: they are all self-trapped 

entities possessing particle-like behavior. 

In this Chapter we outline the mechanisms through which photorefraction 

can support optical spatial solitons (for a specific review, see pages 61-125 

in [4]), give an account of the development of the main underlying ideas, 

and describe the associated phenomenology. Since the discovery of photorefractive 

solitons in 1992 [5], they have become one of the most important 

experimental means to study universal soliton features. The rich diversity of 

photorefractive effects has allowed experimental investigations into a large 

variety of soliton phenomena, many of which have here found their very 

first observation, in any soliton-supporting system in nature. For example, it 

was with photorefractive solitons that fascinating soliton interaction effects, 

such as 3D soliton spiraling, fission and annihilation, were first demonstrated. 

Likewise, random-phase (or incoherent) solitons were first observed in photorefractives, 

and so multi-mode solitons, both in 1D and in 2D. And, very 

recently, solitons in two-dimensional nonlinear photonic lattices were demonstrated, 

once again, in photorefractives, using a real-time optically-induced 

photonic lattice. Today, almost 12 years after their discovery, photorefrac-

2 E. DelRe, M. Segev, D. Christodoulides, B. Crosignani, and G. Salamo 

tive solitons have evolved into a major subject contributing to research at 

the cutting edge of nonlinear science. The intrinsic complexity of the photorefractive 

light-induced index-change, driven by several charge transport 

mechanisms, utilizing linear and quadratic electro-optic effects, and having 

a polarization-dependent tensorial behavior, contributes in giving rise to a 

rich phenomenology. Much is understood now about the formation processes 

of the various types of photorefractive solitons, and many of the parameters 

can be controlled individually: at the same time numerous questions 

are still open. Also important, research on solitons in photorefractives has 

benefited not only the soliton community, but has also introduced a number 

of new ingredients to photorefractive studies at large. For example, understanding 

the propagation dynamics of beams in photorefractives (rather than 

two- and four-wave-mixing), including the formation of self-oscillators (e.g., 

the so-called ”double phase conjugator”) has considerably benefited from the 

understanding gained in photorefrative soliton research. Likewise, exploring 

spatially-localized effects that emerge and find their full realization directly 

within the sample, such as instabilities and spontaneous pattern formation, 

with and without a cavity, is now nicely understood through the intimate 

connection between solitons and modulation instability [4]. These distinguish 

soliton phenomenology from holographic and wave-mixing schemes, which, by 

nature, deal principally with the manipulation of light as a device to an effect 

which occurs outside the crystal. In this Chapter, we attempt to provide an 

updated overview on the fascinating phenomenon of photorefractive solitons, 

which continuously brings in new surprises, new effects, new excitement, to 

the photorefractives community, to the much broader nonlinear optics community, 

and to soliton science at large. 

2 The discovery of solitons in photorefractives 

In the wake of renewed interest in soliton propagation, triggered by studies 

on temporal solitons in an ever more competitive fiber optic network, the beginning 

of the 1990s see an intense effort aimed at finding accessible physical 

systems in which to experimentally investigate spatial solitons. Conventional 

Kerr-type nonlinear schemes presented crippling limitations connected to the 

extremely high optical intensities involved, and suffering from the fundamental 

constraints associated with the instabilities and catastrophic collapse of 

Kerr solitons in bulk media. Motivated by the strong nonlinear response of 

photorefractive crystals, even at low optical intensities, M. Segev, B. Crosignani, 

and A. Yariv, were able to formulate in 1992 the first photorefractionbased 

self-trapping mechanism. This embryonic idea sets the beginning of the 

field, and, indeed, of our description [5]. 

A spatial soliton is a beam which, by virtue of a robust balance between 

diffraction and nonlinearity, does not change its shape during propagation. A 

direct observation of a spatial soliton in photorefractives is shown in Fig.(1).

Photorefractive Solitons 3 

Fig. 1. Top view of spurious scattered light produced by transient centrosymmetric 

solitons in a biased sample of potassium-lithium-tantalate-niobate (KLTN). (a) 

Linear diffraction from an input Full-Width-Half-Maximum (FWHM) of 6 µm to 

150 µm. (b-c) Solitons formed with opposite values of external field. Taken from 

[11] 

Before the discovery of photorefractive solitons, nonlinear optics in photorefractives 

was centered on diffusion-driven wave-mixing schemes, typically resulting 

in high energy exchange between the interacting waves, a mechanism 

at the heart of photorefractive self-oscillation and ”passive” phase conjugators. 

During that phase, other settings, which exhibited elevated phasecoupling 

(and much lower energy-exchange) attracted only remote interest. 

Furthermore, during diffusion-driven photorefractive wave-mixing, beam energy 

is spontaneously driven into modes not present in the launch, leading to 

the highly deteriorating phenomenon called beam fanning: the exact opposite 

of a precursor to the orderly events which accompany a solitary wave. Evidently, 

at the time there was no indication of any kind that photorefractives 

could support solitons. Segev et al., however, argued that since wave-mixing 

was intrinsically accompanied by a mutual phase-modulation, one could find 

a condition in which the plane-wave components of a diffracting beam could 

mix so as to lead to a trapping self-phase modulation: their mutual exchange 

could compensate for the linear dephasing between the plane-wave components 

of a beam, and thus counteract diffraction altogether. They observed 

that, in contrast to schemes where the main mixing agent is diffusion, leading 

to the highly asymmetric fanning process, a symmetric mutual phasemodulation 

could be achieved through the application of an external bias 

field. They concluded that, when the diffusion space-charge field could be neglected 

with respect the external bias field, a symmetric self-focusing occurs 

and self-trapping effects should emerge. Under such conditions, fanning itself, 

being mediated by diffusion, would have a secondary effect [6]. 

In order to achieve a set-up symmetric with respect to the propagating 

axis, the beam should be launched in a zero-cut uniaxial sample along the 

ordinary axis a, with the external biasing field E0 applied along the poling 

optical axis c, through two electrodes brought to a relative potential V .


An optimal arrangement was identified in Stontium-Barium-Niobate (SBN) 

doped with rhodium impurities. Characterized by an r33 220pm/V ,for 

an extraordinary c polarized beam launch, index modulations of the order 

of |∆n| (1/2)n 3 r33E0 ∼ 2 − 5 · 10 −4 for achievable fields of the order of 

E0 ∼ 1-3kV/cm could be reached, a nonlinearity sufficient to support the 

formation of a 10 µm wide soliton. 

Fig. 2. The original soliton observation schematic, as reported in [7]. The extraordinary 

launch beam is focused in proximity of the input facet. Transient dynamics 

were detected recording the output intensity transmitted through an appropriate 

aperture positioned before the output detector. 

The first pioneering experiments, reported in [7] were carried out with 

the set-up indicated in Fig.(2). In relating the first results, we more often 

delve upon details of transient - as opposed to steady-state - effects, and 

miss the main and revolutionary point: where no previous phenomenology 

even hinted at self-focusing, these first experiments indicated, unmistakably, 

that a visible continuous-wave beam propagating in a biased sample would 

actually self-trap, the ensuing spatial soliton being readily accessible to direct 

observation (see Fig.(1)). Although the initial conjecture would later 

not prove to be greatly adherent with first findings, it did express the main 

physical idea supporting the phenomena. These pioneering results established 

that very narrow beams launched in a properly-biased photorefractive crystal 

would self-trap, and propagate in a robust fashion, undistorted by fanning 

and other noise sources in the crystal or even fairly large deviations from 

optimal launch conditions [8]. For example, it was established that a 15 µm 

sized continuous-wave 457nm µW beam would not suffer fanning and selftrap 

for external fields from 400-500 V/cm. This observation led to a rapid 

series of predictions and experiments, which now form the phenomenological 

basis of photorefractive spatial solitons [8, 9, 10]. 

As commonly occurs when scientific progress is in action, those first experiments 

posed more questions than answers. First, the results indicated a transient 

self-trapping process (see insert in Fig.(2)), during a time window of 

several hundreds of milliseconds, not characterized by stringent existence con-


ditions, as would have been expected for normal self-trapping, where diffraction 

is exactly balanced by a specific value of nonlinearity. On the contrary, 

beam intensity and applied bias field, two parameters assuredly implicated 

in the nonlinear response, could be considerably varied without significant 

changes in self-trapping. But even more astonishingly, the tentative launch 

of two-dimensional (circular) beam led to its stable self-trapping. This at 

once indicated that the underlying nonlinear mechanism was not Kerr-like, 

since the catastrophic collapse associated with self-focusing in Kerr media 

did not occur. For some reason, the intrinsic anisotropy of the light-induced 

space-charge-field, resulting both from the application of an external bias 

along one transverse direction only, and the directionally resolved electrooptic 

response, allowed for a two-dimensional soliton [7, 8, 9]. Interestingly, 

both aspects, the transient/non-parametric and quasi-circularly-symmetric 

nature of these first manifestations, which are reproducible in all photorefractive 

materials that can support solitons, are still not fully understood 

even today (see sections 4 and 5). The observation of a two-dimensional soliton 

in a bulk medium has attracted interest and sometimes fomented outright 

debate. At the same time, the transient nature was generally looked upon as 

an undesired and limiting effect. It cast a shadow both as to the nature of 

the interaction, but more importantly, as to its stability. Transient effects of 

the sort had a history in photorefraction, and they were attributed to charge 

accumulation in dark regions of the sample of the photo-excited charge, depleting 

illuminated portions and possibly screening external bias. 

This triggered the idea that the transient nature of the self-trapping was 

an effect of charge accumulation screening E0: free-charges would be photoexcited 

across the beam profile, and, drifting in the external field, would reach 

the bordering dark regions, and get trapped there. These trapped charges 

would give rise to an internal (space-charge) field with a polarization opposite 

to E0. In SBN, this lowering (screening) of the applied field at the illuminated 

regions would locally lead to electro-optic lensing. The decay of this 

(induced) lens with time would then be a consequence of the fact that charge 

would continue to separate until E0 was totally screened, thereby saturating 

and flattening the induced ”lens”. In this, investigators found the solution: 

they would foresee a compensating mechanism through which accumulated 

charge could be eliminated by homogeneously illuminating the sample, which 

amounts to increasing the dark sample conductivity (see Fig.(3)). On the basis 

of the relative intensity of the beam to the background, there would exist 

a dynamic equilibrium leading to a steady-state lensing effect [12, 13, 14, 15]. 

As the model was reformulated on this new, an to some extent, simpler 

screening idea, foreseeing the artificial enhancement of crystal dark conductivity, 

Castillo et al. [16] reported steady-state self-focusing, using a homogeneous 

illumination to free accumulated charge. Finally, Shih et al. [17, 18] and 

Kos et al. [19] were able to not only observe this non-transient soliton phenomenology, 

but also relate it to the new model. This type of self-trapping is


Imaging 

Lens 

CCD 

CCD 

Photorefractive 

Sample 

Spherical 

Lens 

Polarizer BS 

V 

c 

Beam 

Expander 

Fig. 3. Scheme used to stabilize screening solitons, as described in [17] and [19]. 

Now the diffracting launch beam is accompanied by an ordinary unfocused wave 

appropriately extracted from the single laser. 

since termed a screening soliton, and constitutes the most commonly studied 

type of photorefractive soliton. Its explanation requires a treatment which 

goes beyond the small modulation depth treatment, and for which single 

mode description, such as the involved in two-wave mixing, is unsatisfactory. 

Photorefractive solitons have since been observed in SBN, BSO, BGO, 

BTO, BaTiO3, LiNbO3, InP, CdZnTe, KLTN, KNbO3, polymers and organic 

glass. 

3 A saturable nonlinearity 

The formulation of a descriptive and predictive theory for photorefractive 

solitons involves some profoundly different aspects and theoretical tools than 

those employed in traditional wave-mixing theories. First, no periodic structure 

is present, and second, in most configurations, all the physical variables 

vary across the beam profile by a large fraction (e.g., from peak to zero intensity) 

such that the modulation cannot be treated as a small perturbation. 

However, steady-state photorefractive solitons have two intrinsic symmetries 

which reduce the problem: they are evidently time-independent, and their 

intensity I is independent of the propagation coordinate z. Yet the heart of 

complexity is nonlinearity, and even for a z-invariant photoionizing intensity I 

there is still a wide range of parameters, of which only a small subset can support 

solitons. In order to formulate a semi-analytic theory, a one-dimensional 

reduction must be implemented: the beam should be such that no y-dynamics 

emerge, the soliton intensity being solely x-dependent [I(x)]. Experimentally, 

this was achieved by launching a beam focused down through a cylindrical 

lens, and quite similarly this has led to quasi-steady-state self-trapping in the 

absence of background [9], and to steady-state screening solitons for appropriate 

values of E0 (see Fig.(4)) [19]. 

In fact, in many aspects these one-dimensional (stripe) waves, generally 

termed one-plus-one dimensional screening solitons [(1+1D)], share with 

z 

y 

PBS 

x 

Laser


Fig. 4. A one-dimensional soliton observed in biased SBN. Top: input intensity distribution, 

output linear diffraction (no bias), and output self-trapped beam. Center 

and bottom: profiles. As reported in [19] 

their needle-like counterparts, which are therefore two-plus-one-dimensional 

screening solitons [(2+1D)], the same behavior. Although (2+1)D self-trapping 

in photorefractive media is still not fully theoretically understood (see section 

4), the theory of (1+1)D photorefractive solitons [12, 13, 14, 15] constitutes 

the basic confirmation that photorefraction supports self-trapping. 

In most conditions of interest, the optical intensity distribution I(x) is 

such that, for an electron-dominated photorefraction, the resulting concentration 

of photo-excited electrons N, the concentration of acceptor impurities 

Na, and the concentration of donor impurities Nd follow the hierarchy 

N


be identified with a saturation scale, i.e., that spatial scale under which the 

maximum attainable charge (when the concentration of ionized donors N + 

d ≈ 

Na) cannot screen E0 (which becomes comparable with the saturation field 

Eq). From Eq.(1) Y and Q are now related through 

′ 

YQ Q Q 

+ a − 

1+Y ′ 1+Y ′ (1 + Y ′ 

′′ 

Y = G, (2) 

) 2 

with a = NakbT/ɛE2 0 and G = gE0/Ib. The prime stands for (d/dξ). Equation 

(2) can be formally solved, without approximations, for Y to give 

Y = −a Q′ 

Q 

′ 

′′ 

G GY Y 

+ + + a . (3) 

Q Q 1+Y ′ 

We can now identify the various terms with precise physical processes, as we 

shall see. Equation (3) is rendered tractable by the fact that the greater part 

of spatial soliton studies involve the trapping of beams with an intensity Full- 

Width-Half-Maximum (FWHM) ∆x ∼10µm. For most configurations, xq ∼ 

0.1 µm, andη = xq/∆x ∼0.01 represents a smallness parameter. In other 

words, screening solitons do not deplete photorefractive charge. A dimensional 

evaluation of the various terms for the appropriate high-modulation 

regime indicates that 

Y (0) = G 

+ o(η), (4) 

Q 

since a ∼2.5 , and G -1 [15]. A first correction is obtained by iterating this 

solution into Eq.(3), and the resulting expression for Y is 

Y (1) = G Q′ 

− aQ′ − 

Q Q Q 

2 G 

+ o(η 

Q 

2 ). (5) 

The first dominant term is generally referred as the screening term. It represents, 

in our discussion, the main agent leading to solitons. It is local, in that 

it does not involve spatial derivatives or integration, has the same symmetry 

of the optical intensity Q, and represents a decrease in E with respect to 

E0 on consequence of charge rearrangement (G -1). So perhaps the most 

astonishing fact of our discussion is that, in truth, for a large variety of conditions, 

this form of self-focusing (including self-defocusing) is the dominant 

effect, as the plentiful family of reported observations that have followed the 

1992-1993 discovery imply. The second term, of first order in η, is simply 

the high-modulation version of what is generally called the diffusion field 

(Ed). The third, again of first order in η, is the coupling of the diffusion field 

with the screening field, a component sometimes referred to as deriving from 

charge-displacement [21]. Both these two last terms are nonlocal, in that they 

involve a spatial derivative, and thus provide an anti-symmetric contribution 

to the space charge field (Y ) for a symmetric beam I(x) =I(−x). That is, 

these last two terms lead to a beam self-action of opposite symmetry of that


required to support solitons: whereas for conventional mixing studies they 

play the central role, here they lead to beam self-bending (or self-deflection), 

which, for most configurations, amounts to a slight parabolic distortion of the 

preferentially z-oriented trajectory. The subject has attracted interest over 

the years and has helped build an understanding into the limits of the local 

saturable nonlinearity model [10, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29]. 

In order to identify the nonlinearity, we must now translate the spacecharge 

field E into an index modulation. The standard soliton set-up is such 

that a negative uniaxial zero-cut sample is positioned so that the x-axis is 

the direction along which E0 is applied, the soliton beam of intensity I is 

extraordinarily-polarized and is propagating along z, while Ib is obtained 

through a co-propagating ordinarily-polarized plane-wave [17]. For a noncentrosymmetric 

photorefractive, like SBN, ∆n = − 1 

2n3r33E, n being the 

unperturbed crystal index of refraction, and rij the linear electro-optic tensor 

of the sample. On taking, consistently with our iterative scheme, the 

expression of Eq.(4), we obtain the nonlinearity 

∆n(I) =− 1 

2 n3 V 1 

1 

r33 

= −∆n0 , (6) 

Lx 1+I/Ib 1+I/Ib 

which constitutes a saturable nonlinearity, identical (within a constant term) 

to the nonlinear index change in a homogeneously-broadened two-levelsystem. 

3.2 The soliton-supporting nonlinear equation 

A soliton is loosely defined as a wave that preserves its shape and velocity 

throughout propagation, while, very importantly, displaying a particle-like 

behavior when made to interact (”collide”) with other solitons. As such, solitons 

possess a number of conserved quantities, such as power, momentum, 

Hamiltonian, etc. [4]. Optical spatial solitons, in their scalar manifestation, 

are generally governed by the nonlinear equation for a monochromatic paraxial 

beam 

 

∂ i 

− 

∂z 2k 

d2 dx2 

A(x, z) =− ik 

∆nA(x, z) (7) 

n 

where k =2πn/λ is the wave-vector, A is the extraordinary component of 

the slowly varying optical field, i.e. Eopt(x, z, t) = A(x, z)exp(ikz − iωt), 

ω =2πc/nλ, andI = |A(x, z)| 2 . We seek stationary (non-diffracting) solutions 

of the form A(x, z) = u(x)eiΓ z√Ib, normalize the transverse spatial 

scale to the so-called nonlinear length scale d = (±2kb) −1/2 , i.e., 

ξ = x/d, which for photorefractive solitons is obtained from the expression 

b =(1/2)kn2r33(V/Lx), and obtain [12, 15] 

d 2 u(ξ) 

dξ 2 

= ± 

Γ 

b − 

1 

1+u(ξ) 2 

 

u(ξ). (8)


The plus sign is for b>0, the minus for b0, or a self-defocusing, 

for b0, and we observe a self-focusing 

nonlinearity. It is possible to apply E0 in a direction opposite to ferroelectric 

axis, thus effectively changing the sign of b, then E0 must be smaller than the 

coercive field, otherwise it may render the ferroelectric crystalline structure 

unstable and de-pole the crystal. Both defocusing and focusing nonlinearities 

support solitons. A self-focusing nonlinearity traps a conventional bell-shaped 

beam into a bright soliton; a self-defocusing nonlinearity traps a wave feature, 

such as a notch generated by a π phase jump, whose structure is enlarged by 

the diffraction of surrounding illuminated regions, the ensuing propagation 

invariant wave being termed a dark soliton. 

Equation (8) can be integrated (by quadrature) once, giving the relationship 

Γ/b = log 1+u2 

2 

0 /u0 for bright beams, and Γ/b =1/ 1+u2 

∞ for 

dark, where u∞ = u(∞) =−u(−∞), and u0 = u(0) (u2 0 = I(0)/Ib being 

referred to as the intensity ratio). 

3.3 Soliton waveforms and existence curve 

As can be imagined, the self-trapped waves u, solutions of Eq.(8), form an isolated 

subset of all possible dynamical evolutions described by Eq.(7). The imposition 

of z-invariance implies not only a specific relationship between beam 

parameters, but fixes the actual waveform u in all its details (see Fig.(5)). In 

general, such a prediction can have little if no physical meaning. For solitons, 

however, we have two countering effects, diffraction and self-focusing, which 

are coupled by nonlinearity to form a feedback mechanism. The result is that 

most soliton solutions are stable to perturbations, and represent an attractor 

to system dynamics. This, in turn, contains the beauty and physical appeal of 

soliton physics: that a propagating beam, interacting in a non-trivial way with 

a hosting medium, should preferably be attracted to a robust, propagationinvariant, 

and very specific wave-form, instead of producing an unstable and 

unpredictable chaotic system. 

Returning to our system, what are the soliton wave-forms, and, more 

importantly, what are the beam parameters, ∆ξ (associated to ∆x) and intensity 

ratio u 2 0, that characterize the subset of soliton solutions? The issue 

is of particular importance, because, although self-trapping forms an attractor, 

launch experiments are designed to deterministically lead to a soliton, 

a scheme that requires the launch to be close enough to a self-trapped wave 

in parameter space (intensity u 2 0 and normalized width ∆ξ). For the family 

of integrable nonlinear equations, such as the Sine-Gordon, the Nonlinear 

Schroedinger, and the Korteweg and de-Vries equations, an explicit solution


can be found, for which the relationship between beam ∆ξ and u 2 0 is unique. 

However, for the saturable nonlinearity described by Eqs.(6,8) the ∆ξ versus 

u 2 0 relation is not unique, but instead yields a continuous curve commonly 

termed the soliton existence curve [15] (see Fig. (6)). 

Fig. 5. Soliton waveforms for bright solitons, as reported in [19]. 

The wavefunctions of bright photorefractive screening solitons are bell-shaped 

functions which are neither a Gaussian nor a hyperbolic secant [30]. Experimentally, 

such solitons are generated by launching a focused-down onedimensional 

Gaussian beam with a ∆x and peak intensity I0 such that, for 

the given configuration of crystal parameters n, r33, andIb, for the given bias 

E0, the resulting values of ∆ξ and u0 lie on the existence curve. For dark 

solitons, in turn, the same procedure can be implemented for the relevant 

(u∞,∆ξ) parameter space. 

3.4 Experiments and theory 

The main advantage of having formulated the theory highlighting the saturable 

nature of the nonlinearity is that, within the limits in which the underlying 

approximations are valid, it allows the prediction of solitons as a 

specific feature independent on the particular experimental configuration. 

Thus a physically identical soliton will emerge for two self-trapped beams 

of, say 10 and 20 µm, as long as the applied voltage V , material response,


Fig. 6. Soliton bright and dark existence curve, as reported in [19]. 

beam intensity and background illumination are such as to project the two 

conditions on the very same point on the (u0, ∆ξ) parameter space. Note 

that this powerful device breaks down as soon as we consider the o(η) terms, 

or even simple corrections in the actual nonlinearity, such as those deriving 

from dielectric nonlinearity [31]. 

In order to test the validity of the treatment, experiments have been carried 

out with the aim of detecting for what conditions of launch a 1+1D 

steady-state soliton would form [19, 32, 33], and some examples of results are 

contained in Fig.(7) for bright solitons and Fig.(8) for dark. 

Indeed other experiments have led to much the same conclusions, for 

which the qualitative agreement is full, but for some regions of parameters, 

where the quantitative comparison is weaker. At present it is believed that 

the discrepancy is due more to the presence of extraneous effects than to 

a deficiency in the model. For example, it has been observed that some of 

the background illumination, which is made to propagate along an ordinary 

mode, in order to not undergo substantial evolution, is actually trapped by 

the soliton through the finite r13 coefficient. Another source of uncertainty 

is connected to the difficulty in establishing the precise value of rij for the 

given sample: this can depend on the level of purity of the poling, on the 

presence of considerable clamping and on temperature. 

Concerning more fundamental aspects of the model, we note that although 

the results shown in Fig.(7) support the approximation contained in Eq.(4), 

the actual beam evolution shows a clear and reproducible self-bending effect


Fig. 7. Comparison between experiments and theory for (1+1)D bright screening 

solitons, from [32]. Here the low-intensity regime is what we specifically term 

screening solitons. 

Fig. 8. Comparison between experiments and theory for (1+1)D dark screening 

solitons, from [?]


that can only find its explanation in the full expression of Eq.(5). It appears 

that, even though these nonlocal terms produce a parabolic trajectory, they 

do not greatly influence the existence curve. From a different perspective, 

we note that self-bending becomes an important issue when experiments are 

carried in highly solitonic regimes, characterized by a large ratio between selftrapped 

propagation distance and linear diffraction length Lz/Ld: to increase 

Lz/Ld we can either use a longer sample, this increasing nonlinearly the 

lateral shift, or use tighter launch beams, this increasing η. 

4 Two-dimensional solitons 

The (1+1)D screening solitons represent the firm experimental and theoretical 

foothold on which a large part of research rests, especially because of the 

appealing nature of the saturable nonlinearity which gives rise to numerous 

(fascinating) soliton interactions not present with Kerr-type solitons. However, 

the most ground-breaking achievement, both from the physical and from 

the applicative point of view, is the self-trapping of (2+1)D solitons. As their 

lower-dimensional counterparts, needles form both in the transient regime 

- as quasi-steady-state self-trapping [7] - and in temporal steady state as 

(2+1)D screening solitons [17, 18]. Such needle-like solitons were originally 

documented in SBN, and have been replicated in most soliton-supporting 

photorefractive media, such as other ferroelectrics [34], semiconductors [35], 

paraelectrics [36], sillenites [37], and indeed for most types of self-trapping: 

bright, photovoltaic [38], multimode [39, 40, 41], and incoherent [42, 43], to 

name a few. Furthermore, even (2+1)D dark solitons were observed in photorefractives, 

in quasi-steady-state [44] and in steady-state [45] under a bias 

field, as well as photovoltaic [46] and incoherent [47] dark ”vortex” solitons. 

An example of a dark vortex screening soliton is shown in Fig.(12). Twoplus-one 

dimensional solitons form when a circularly symmetric beam is focused 

down onto the input face of the sample, and the initially diffracting 

beam collapses into a non-diffracting 2D beam having an almost ideally 

circularly-symmetric shape (see Fig.(9)). These studies, which constitute 

one of the rare possibilities of observing (2+1)D solitons, have greatly 

contributed to the understanding of the physics associated with higher-thanone-dimensional 

nonlinear waves. The observation of soliton spiraling in full 

3D [48], as well as fusion, fission, birth and annihilation of (2+1)D solitons 

[49, 50, 51], have extended the very concept of soliton-particle behavior. 

The description and understanding of the mechanisms that support 

(2+1)D solitons are, to some extent even today, still incomplete. This is 

because the propagation involves a nontrivial three-dimensional, anisotropic, 

and spatially nonlocal nonlinear problem [52, 53, 54, 55, 56, 57]. The fact 

that photorefractives can support self-trapped needles of (almost ideally) 

circularly-symmetric shape seems very surprising right from the outset. More


Fig. 9. The direct and detailed observation of a circularly-symmetric 12 µm needle 

phenomenology in a non-zero-cut sample of SBN, from [18]. 

specifically, the external field is applied between two parallel planar electrodes, 

and thus breaks the circular symmetry of the problem. The explanation 

relates to the fact that the (space charge) field lines bend in the regions of 

higher illumination, and, for some range of parameters seem to yield a quasiradial 

distribution of the field component giving rise to a nonlinear index 

change. For example, in SBN this means that the c-component of the space 

charge field has roughly a circular symmetry. The very fact that such (2+1)D 

propagate in a stable fashion, not undergoing catastrophic collapse (as such 

solitons in Kerr media would), is a direct indicative that the photorefractive 

nonlinearity is saturable also in two (transverse) dimensions. However, during 

the temporal transients, and for various values of applied field, self-trapping 

manifests considerable beam ellipticity [52], excluding the direct validity of 

a circularly-symmetric 2D saturable model. Nevertheless, the large amount 

of experimental evidence on 2D solitons in almost every photorefractive material 

in which solitons have been identified, implies that a modified model, 

possibly anisotropic and slightly nonlocal, should exist [55]. What clears the 

picture are the studies on (2+1)D soliton interaction-collisions. On the one 

hand, spiraling and large angle collisions show that whatever anisotropic components 

emerge, they do remain localized around the beam [48, 58], whereas 

lower angle collisions indicate unmistakably the presence of a saturable yet 

anisotropic nonlinear behavior: the repulsion of incoherent needles [59].


From a theoretical perspective, the system has two fundamental anisotropies: 

the boundary conditions, which emerge from the application of an external 

bias along the x-direction, and the electro-optic response, which implies a 

complex tensorial index modulation depending on the beam polarization, direction 

of the local electric field E, and the relative orientation with respect 

to the crystal lattice. The result is that the nonlinear response has a nonlocal 

component that is superimposed on the saturable component [54]. In order 

for a quasi-circular optical symmetry to appear, the underlying space-charge 

field E must be anisotropic, manifesting two characteristic lateral lobes (see 

Fig.(10)) [60]. The appearance of these features, that have an increased complexity 

with respect to the ionizing intensity I(x, y), are the basic manifestation 

of a nonlocal mechanism. Contrary to what might seem logical, if one 

attempts to identify their origin extending our knowledge of (1+1)D solitons, 

these lobes do not emerge because of the nonlocal mechanisms indicated in 

Eq.(5). These lobes emerge due to the matching of boundary conditions in 

the higher dimensional case, i.e., they match the local field structure, in the 

beam cross-section, to the x-oriented E = E0 far from the beam. 

In the 2+1D case we start from the basic relation, valid to zero order in 

η, 

and the irrotational condition 

∇·(Y Q) = 0 (9) 

∇×Y =0. (10) 

From these a lobular structure emerges (see Fig.(11)). This anisotropy which 

is not at all connected to η, is simply not small in any accessible case. In 

other words, the nonlocality in the (2+1)D case is intrinsic to the process, 

whereas it is merely a correction for (1+1)D solitons. 

The fact that 2+1D solitons are supported by this more complex nonlinearity 

does not substantially modify our soliton picture. One burdening 

consequence, however, is that we do not have a means to formulate in a 

straightforward manner an existence curve for (2+1)D solitons, nor can we 

unequivocally say if such a concise and powerful tool even exists for the 

higher dimensional solitons. Nevertheless, if we phenomenologically build the 

set of points in which it is possible to observe circular-symmetric self-trapping 

[18, 36], we find a single valued continuous curve that behaves and looks just 

like the existence curve of (1+1)D solitons (albeit at somewhat higher values 

of ∆ξ). 

Evidence on both the existence of circularly-symmetric solitons and of 

their intrinsic difference from (1+1)D solitons in photorefractives is highlighted 

by the experimental studies on transverse instability of (1+1)D solitons 

in bulk media. In those experiments, increasing the nonlinearity leads 

a (1+1)D soliton (at the proper value of nonlinearity vs intensity ratio, as


Fig. 10. Needle electro-holography from [60]. (a-c) conventional formation of a 

centrosymmetric soliton; (d) electroholography showing the lobes. 

Fig. 11. Numerically evaluated x-component of E (top) for the soliton beam profile 

(bottom), from [55].


Fig. 12. A screening vortex and its guiding features, from [45]. (a) Input intensity 

distribution of the vortex; (b) Diffracting vortex after linear propagation to the 

output of the sample; (c) Self-trapped output intensity distribution in a biased 

sample. (bottom) Probe beam guided propagation. 

Fig. 13. Transition from (1+1)D to (2+1)D self-trapping, from [61]. 

determined by the existence curve); then, a further increase in the nonlinearity 

results in beam breakup into an array of circularly-symmetric solitons, as 

shown in Fig.(13) [61]. 

5 Temporal effects and quasi-steady-state dynamics 

One of the main characteristics of photorefraction is that it is a cumulative 

process. This is the origin of its elevated response even at low optical intensities. 

Evolution is dominated by a charge redistribution process which 

leads to a time constant τ ∝ 1/I. For small-modulation depth conditions I is 

almost constant, and a single-time-constant behavior can be identified. For 

spatial solitons the story is quite different: not only are we faced with a high 

modulation response, and the time constant changes in the transverse plane, 

but since they form out of a three-dimensionally resolved diffracting beam, 

time behavior is also resolved in the propagation direction. The result is an 

evolution which presents a number of surprising phenomenologies, which are,


however, complicated and difficult to describe (an example that goes beyond 

soliton studies can be found in [62]). 

Possibly the most important, and, indeed, surprisingly, effect is observed 

when launching a diffracting beam in an appropriately biased sample, without 

any appreciable background illumination. As reported in the very first 

experiments on solitons by Duree et al. in Ref.[7], the beam is observed to 

form, after a time interval of the order of 10ms, a spatial soliton, remain 

almost stationary for an interval of 20ms, and then decay into a once again 

diffracting beam. This peculiar sequence, which involves a rare plateau evolution, 

is referred to as a quasi-steady-state soliton. 

The time dependent version of Eq.(2) truncated at zero order in η reads 

∂Y 

+ QY = G, (11) 

∂τ 

where τ = t/τd, τd = ɛ0ɛrγNa/(qµs(Nd−Na)Ib) is the characteristic dielectric 

time constant, γ is the recombination rate, µ the electron mobility, s the donor 

impurity photoionization efficiency, and, as occurs for most configurations of 

interest to soliton dynamics, the charge recombination time τr=1/(Nγ)is 

much shorter than charge displacement dynamics, and no time dependence 

in the boundary conditions are considered (G=-1). For small modulation, 

Eq.(11) gives the characteristic exponential dynamics with t = τd/Q, but as 

soon as Q is spatially resolved, and furthermore is itself evolving, a continuum 

of different time constant contribute. Seen in a different perspective, soliton 

time evolution is highly time-nonlocal, as the formally equivalent integral 

version of Eq.(11) 

Y = e − 

τ 

τ 

′ 

Qdτ 

0 1+ dτ 

0 

′ τ 

′ 

′′ 

Qdτ 

e 0 , (12) 

indicates [54]. The full complexity of this behavior emerges during transients, 

i.e., when I changes in an appreciable manner with time. This occurs most 

evidently during the very first collapsing stage, for times τ ≤ 1/(1 + u2 0), and 

leads to a stretched exponential evolution [63]. A characteristic of multi-scale 

processes, this behavior is common to the entire family of soliton supporting 

cumulative nonlinearities. A numerical approach to Eq.(12) coupled to the 

parabolic wave equation confirms experimental findings, but to date there is 

no clear understanding as to why the soliton should pass through a plateau, 

and, more importantly, how to evaluate the so-called threshold nonlinearity 

for which self-trapping is achieved, the duration of the plateau, and the 

nonlinear equation, such as Eq.(8), for which the waves are eigenfunctions. 

Furthermore, we do not have a means to predict the actual trapping ∆x at 

the plateau, for a given nonlinearity. 

In order to highlight the sole role of high modulation depth, we can considerably 

simplify our predictions, excluding nonlocality, by imposing that Q 

not have relevant time-dependence [64]. In this case Eq.(12) is simplified to 

give


Y = e −τQ (1 + 1 

Q (eτQ − 1)), (13) 

where Q is an appropriate time average. Whereas this approach can be 

meaningful and useful for conditions in which a soliton (i.e., the beam I) 

is steady, such as for steady-state incoherent solitons which we will describe 

below, it can say nothing as to beam transients. It has been speculated that 

Eq.(13) could be valid when a negligible amount of diffraction is involved 

[65, 66, 67, 68, 69]. 

Delaying the discussion of incoherent self-trapping, a consequence of cumulative 

inertial response, to section 10, we add a mention to the wealth of 

transient phenomenologies that occur for higher-dimensional needles [70], and 

those associated to a time-dependent external bias E0 [11, 71, 72, 73]. Lastly, 

we should mention the study of single pulse propagation and space-charge 

build-up [74, 75]. 

6 Various photorefractive mechanisms supporting 

self-trapping 

One of the nicest features of self-trapping of optical beams in photorefractives 

is the diversity of mechanisms that can support solitons. Apart from 

the solitons described in the previous section, which relied on an externallyapplied 

bias field, self-trapping in photorefractives can also arise from photovoltaic 

effects [76], or from diffusion-driven effects [77], or from resonantlyenhanced 

effects caused by the excitation of both electrons and holes [78]. In 

several cases, combinations of two of these effects can also lead to solitons 

[e.g., solitons supported by the photovoltaic and the screening nonlinearities 

simultaneously]. Furthermore, in some cases, self-trapping can arise from 

semi-permanent changes in the crystalline structure, either though clustering 

of ferroelectric domains [79], or through re-poling of macroscopic regions 

[80, 81], both being driven by the local space charge field. Such permanent 

changes are in fact ”fixed” (soliton-induced) waveguides, acting as microstructure 

optical circuits ”impressed” into the volume of the bulk nonlinear crystal. 

To this date, this is one of the very few techniques to create intricate 3D optical 

circuitry. In this section, we briefly review these additional mechanisms 

supporting self-trapping of optical beams in photorefractive media. 

6.1 Photovoltaic solitons 

Soliton-supporting mechanisms appear in photorefractives also in the absence 

of electric fields, the major example being photovoltaic solitons [76, 82]. Here, 

in open-circuit conditions and for the (1+1)D geometry, the non-uniform optical 

excitation translates into a non-uniform photoinduced current. This, at


steady state, must be countered by the drift of photo-excited charge (electrons) 

in response to E. Under conditions analogous to those leading to 

Eq.(4), for a beam with features ∆x of the order of several microns, we 

arrive again at a saturable nonlinearity 

∆n(I) =− 1 

2 n3 I 

I 

r33Ep = −∆n0,p , (14) 

Id + I Id + I 

where Ep = βphNaγ/(qµs), Id the equivalent dark illumination, the constitutive 

relation for the current along x being J = qµNE + βph(Nd − N + 

d ). Much 

in the same fashion of Eq.(8), this nonlinearity leads to a nonlinear soliton 

equation that supports bright and dark solitons on the basis of the sign of 

∆n0,p, i.e., on the sign of βph [76, 82]. 

Most of the experiments with photovoltaic solitons have been carried out 

in LiNbO3, for which βph is negative for an extraordinary beam propagating 

along z, this leading to the observation of one-dimensional dark photovoltaic 

solitons [83]. 

Fig. 14. Photovoltaic dark soliton setup, from [83]. Note the use of a resolved 

transverse phase-structure obtained by having part of the launch beam pass through 

an appropriate piece of glass. 

As occurs for the screening type nonlinearity, photovoltaic solitons can 

also form in the higher-dimensional case. For LiNbO3, theseformasdark 

vortex solitons which are supported by a ”spiraling” transverse phase modulation 

[46] (see Fig.(16) ). In KNSBN, photovoltaic self-action is self-focusing, 

and even bright (2+1)D photovoltaic solitons have been detected [38]. Moreover, 

it has been predicted and demonstrated that the use of a background 

illumination, not a strict requirement in photovoltaics, allows the transitions 

from the defocusing to a focusing nonlinearity in LiNbO3 [84]. 

6.2 Resonantly-enhanced self-trapping in semiconductors 

One of the nicest features of photorefractive solitons is the very low power 

level at which they can form, allowing soliton experiments with microwatt


Fig. 15. Observations of dark photovoltaic solitons, from [83]. (a) linearly diffracting 

intensity distribution of a dark notch; (b) self-trapped intensity distribution, 

for various propagation distances in the sample. 

Fig. 16. Self-trapping of a photovoltaic vortex, from [46]. (a) output intensity 

distribution before self-focusing begins, and phase pattern; (b) output intensity 

distribution trapped by the photovoltaic field, and phase pattern. 

(and lower) power levels. As will be discussed in the section on applications, 

photorefractive solitons, and the waveguides they induce, combine properties 

that suggest interesting applications ranging from reconfigurable directional 

couplers, beam splitters, waveguide switching devices, tunable waveguides for 

second harmonic generation, and highly efficient optical parametric oscillators 

in soliton-induced waveguides. In general, however, the formation time 

of solitons in most photorefractive materials is rather long, except when very 

high intensities are used [32]. This is because the photorefractive nonlinearity 

relies on charge separation, for which the response time is the dielectric 

relaxation time, i.e., inversely proportional to the product of the mobility 

and the optical intensity, and the mobility in photorefractive oxides is low. 

In principle, however, photorefractive semiconductors, (e.g., InP, CdZnTe), 

have a high mobility and could offer formation times a thousand fold faster 

than in the other photorefractives. However, the electrooptic effects in these 

semiconductors are tiny, which implies that solitons that are as narrow as 20 

optical wavelengths necessitate very large applied fields, making solitons in


them almost impossible to observe. But, in some of these materials (InP and 

CdZnTe) that have both holes and electrons as charge carriers, a unique resonance 

mechanism can greatly enhance the space charge field by as much as 

10 times (and more) over the applied electric field. This enhancement yields 

large enough self-focusing effects that can support narrow spatial solitons. 

The resonant enhancement of the space charge field has led to the observation 

of solitons in photorefractive InP [78, 35] and CdZnTe [85]. 

As mentioned, the resonant enhancement of the space charge field occurs 

in materials with both types of charge carriers, both being excited from a 

common trap level: one excited optically and the other excited by temperature 

(or by a second optical beam of a longer wavelength). These two excitations 

work in opposing fashions: one fills (populates) the mid-gap traps whereas 

the other empties them. At steady state, when a focused beam illuminates 

a biased crystal of this kind, and the beam intensity is such that the photoexcitation 

rate of one type of carrier is comparable to the thermal excitation 

rate of the other type of carrier, the concentration of both free carriers at 

the illuminated region decrease drastically. The intuitive explanation is as 

follows [86]. Under proper conditions, the ratio between the concentrations 

of electrons and holes is equal to their ratio in the absence of light, and thus 

has a constant (coordinate-independent) value (see appendix of [17]). The 

net excitation rate of the traps is the difference between the thermal (holes) 

and optical (electrons) excitation rates. At resonance, the net excitation rate 

goes to zero. At the same time, at steady state the excitation rate must be 

equal to the recombination rate, which, in turn, is proportional to the free 

charge concentration. Hence, at resonance (when the excitation rates of holes 

and electrons are comparable) the free charge concentration goes to zero. 

Consequently, the local electric field is highly enhanced because the current 

at steady state must remain constant throughout the crystal. For a given 

temperature, this enhancement occurs at a specific intensity (the resonance 

intensity), for which the thermal and optical excitation rates are comparable. 

It is a resonant enhancement, although it is an intensity-resonance and not an 

atomic resonance. The enhanced electric field compensates for the smallness 

of the electrooptic coefficient and enables a sufficiently large change in the 

refractive index to support narrow solitons. 

The observation of solitons in photorefractive semiconductors [[78, 35, 85]] 

is especially important for several reasons. First, the solitons are generated at 

optical telecommunications wavelengths. Second, they facilitate microsecond 

soliton formation times even at very low (microwatt) optical power. These 

suggest that optical spatial solitons could form from light beams emerging 

from ordinary optical fibers (conventional data-transmission lines) on 

nanoseconds time scales, and could be all-optically switched on-off by employing 

the intensity resonance in photorefractive semiconductors.


Fig. 17. Observed transverse intensity distribution for (1+1)D self-trapping in InP, 

from [78]. 

6.3 Diffusion-driven self-action 

In a conventional noncentrosymmetric crystalline phase, such as that characterizing 

SBN or BaTiO3 at room temperature, charge diffusion leads to an 

asymmetric index profile, which translates into a transverse phase chirp that 

produces self-bending. As discussed in section 3.1 in conjunction with Eq.(5), 

diffusion is typically merely a correction for screening soliton studies. 

Consider now a situation where no external field is applied. On the basis 

of Eq.(1) with g = 0 (null current), E = − kbT 1 dI 

q I+Ib dx . Higher order corrections 

due to saturation in this case are even less important, such that for a 10 

µm beam, they represent a relative import of the order of ɛr ·10−6 , where ɛr is 

the relative dielectric constant. For a sample heated above the ferroelectricparaelectric 

phase-transition, manifesting a quadratic electro-optic response, 

the resulting nonlinearity leads to a symmetric lensing effect, of the type 

1 dI 

∆n(I) ∝ ( I+Id dx )2 . Although in most conditions, such self-action is negligi- 

ble, in the very proximity of the phase-transition, where ɛr attains values of 

the order of 10 4 , self-focusing, the precursor of soliton formation, has been 

observed [77, 87]. The resulting nonlinear equation, which can be extended 

also to the full (2+1)D case, represents the singular situation in which a 

nonlocal nonlinearity (involving a spatial derivative) allows for the explicit 

analytical prediction of both the observed nonlinear diffraction, along with 

such novel effects as ellipticity recovery (see Fig.(18)), and the prediction of 

a full family of solitons, which, however, requiring extremely pure samples 

and precise thermal conditions, and have not been observed. 

6.4 Fixing the photorefractive soliton: Self-trapping by altering 

the crystalline structure 

Solitons in photorefractives are typically supported by the linear polarization 

response to the local space-charge field, P = ɛE. However, optical beams can 

also self-trap in photorefractiue media by altering the crystalline structure


Fig. 18. Diffusion-driven ellipticity recovery. The input ellipticity (a) Λ is recovered 

at the output (b-e) as the crystal temperature T is brought closer to the Curie 

temperature, enhancing diffusion-driven effects, from [87]. 

of the nonlinear medium in which the beam propagates. This happens when 

the local space charge field E becomes comparable with the coercive field 

Ec, and is directed in a direction different from the poling direction. For 

low-modulation gratings, such conditions rarely emerge. But photorefractive 

solitons, being entities with an inherently high modulation depth, are always 

associated with locally high electric fields that can readily depolarize a sample. 

Consider a screening soliton, which exists as a consequence of an external 

bias E0 directed along the poling axis x. Once the soliton has formed, charge 

has redistributed so as to screen, at least in part, the field. This means that 

across the beam profile, charge distribution engenders a field that is approximately 

opposite the bias. Removing the illumination (and the background) 

and switching the external bias off unravels this field. The result is that 

EV =0 −E0 in the region that before hosted the soliton. In SBN Klotz et 

al. have shown how this field can, not only depolarize the sample, but actually 

permanently fix the waveguiding structure that accompanies the soliton, 

an achievement that can have considerable impact in soliton based devices 

[80, 81]. 

Fig. 19. Domain and cluster structure induced by a spontaneous self-trapped process 

in metastable KLTN, from [79]. (a-b) Underlying domain structure as seen 

with the two principal polarizations; (c-d) melting away of domains into clusters, 

clusters into a homogeneous structure, as temperature is increased from the Curie 

point.


Ferroelectric clusters can also play a more dynamic role in spatial selftrapping, 

when they interplay with light-generated space-charge during soliton 

formation. This has been observed in a metastable paraelectric, in proximity 

of the transition temperature, where charge-diffusion fields become 

comparable to Ec [79]. The result is a complicated optical-domain dynamic 

which also leads, in appropriate conditions, to a form of spatial self-action 

known as spontaneous self-trapping (see Fig.(19)). The description of these 

processes, which requires the modelling of domain formation, and their lightmatter 

interaction throughout propagation, is beyond the linear polarization 

approach. Such a theory has yet to be formulated. Nevertheless, the phenomenology 

seems to indicate a more general behavior of propagation in 

metastable hosts. 

7 Alternative photorefractive materials 

Research into nonlinear beam propagation in photorefractives is carried out, 

in parallel, in a series of relatively different materials, each presenting a 

slightly modified version of the mainstream screening type nonlinearity, with 

its advantages and setbacks. Amongst these, we should mention the sillenites, 

such as BTO, BSO and BGO, paraelectrics, such as KLTN, and photorefractive 

polymers [88, 89] and organic gels [90]. 

Some of the original research in photorefractive self-trapping was carried out 

using nonferroelectric sillenites, in a configuration similar to that used for 

ferroelectric samples [16, 91, 92, 93]. The main difference between the physical 

processes is that sillenites present a fairly strong natural optical activity, 

which leads to polarization rotation (during propagation) of both the selffocused 

and the background beams. This causes the effective nonlinearity to 

change in z. Strictly speaking, solitons (diffraction-free) beams cannot exists 

in such optically-active materials [94, 95, 96], at least not in the typical 

scheme of screening solitons. However, using very short samples (a typical 

value of optical activity is ρ0 6 ◦ /mm), and employing pre-compensation 

(in which the beams are not extraordinarily- and ordinarily-polarized, but 

evolve to these halfway through the sample), does allow the observation of 

quasi-solitons. Another setback in using sillenite crystals is in the relatively 

weak effective electro-optic coefficient reff 0.06pm/V (for BTO), which 

implies the use of very high bias fields. Altogether, the sillenite crystals have 

served an important role in the first explorations of steady-state self-focusing 

[16], but have been rarely used for solitons experiments since then, simply 

because other materials offer more favorable conditions for soliton generation. 

Part of screening soliton studies are carried out in paraelectrics, where the 

quadratic electro-optic effect supports a screening nonlinearity in all similar to 

the noncentrosymmetric counterpart, of the form ∆n ∝ (E0) 2 /(1+I/Ib) 2 [97]. 

In these, almost all the conventional photorefractive soliton phenomenology 

can be observed, from (1+1)D [33] to (2+1)D solitons [36]. Dark solitons have


yet to find an appropriate material, because in such crystals the nature of the 

index change cannot be reversed upon reversal of the applied field, as is the 

case for noncentrosymmetric crystals with the linear electrooptic effect. Given 

the generally weak electro-optic response in the high-symmetry phase, most 

studies are carried out in proximity of the ferroelectric-paraelectric transition: 

the crystals must be thermalized to stabilize their dielectric response. 

8 Soliton interaction-collisions 

Nonlinearity generally allows the coupling and energy exchange among beams 

and modes. For solitons, nonlinearity not only supports their propagationinvariant 

nature, but leads to a unique coupling dynamic in which the single 

solitons behave as quasi-rigid particles when they are made to interact with 

one another. In fact, this particle-like dynamic is the very reason for the 

term ”soliton”. In this spirit, interactions between solitons are commonly 

referred to as soliton collisions, and constitute the most fascinating features 

of soliton phenomena. Soliton interactions can be generally classified 

into coherent and incoherent interactions. Coherent interactions of solitons 

occur when the nonlinear medium can respond to interference effects 

that take place when the beams are overlapped. They occur for all nonlinearities 

with an instantaneous (or extremely fast) time response (e.g., the 

optical Kerr effect and the quadratic nonlinearity). For all other nonlinearities 

that have a fairly long response time (e.g., photorefractive and thermal), 

the relative phase between the interacting beams must be kept stationary 

on a time scale much longer than the response time of the medium. When 

this occurs, the material responds to interference between the overlapping 

beams and they exert on each other attractive or repulsive forces, depending 

on the relative phase between the beams. Incoherent interactions, on 

the other hand, occur when the relative phase between the (soliton) beams 

varies much faster than the response time of the medium. In this case the 

resultant force between such bright solitons is always attractive. Altogether, 

soliton interaction-collisions are universal, exhibiting the same basic features 

in spite of the widely diverse physical origins for the self-trapping. For a 

detailed review on soliton interaction forces, see [3]. 

Photorefractive solitons play an especially important role in the study of 

soliton interactions, and have here greatly contributed to soliton research at 

large. This leading role is a consequence of a series of factors: the relative 

ease in soliton generation, which lowers the complexity of multiple soliton 

supporting schemes; the saturable nature of the nonlinearity, which offers 

many features that are simply non-existent with ideal Kerr-type solitons; the 

availability of both (1+1)D and (2+1)D solitons, in a 3D bulk environment; 

and the relatively slow response of photorefractive materials which facilitates 

the possibility of studying both coherent and incoherent collisions in the 

same material system. Over the past eight years, many groups have come


to realize the wealth of possibilities offered by soliton interaction studies in 

photorefractives, and consequently many of the soliton collision features were 

first demonstrated with photorefractive solitons. 

The physical intuition behind soliton collisions relies on the generic idea 

that a soliton is a bound state of its own induced potential, or, in optics, 

a guided mode of its own induced waveguide. Having this in mind, one can 

understand soliton collisions by comparing the collision angle to the (complementary) 

critical angle for guidance in the single soliton-induced waveguides 

(that is, to the angle with the propagation axis below which total internal 

reflection occurs). Whether or not energy is coupled from one soliton into the 

waveguide induced by the other soliton, depends on the relation between the 

collision angle and that critical angle. In terms of a ”potential well”, capture 

depends on whether the kinetic energy of the colliding wave-packets results 

in a velocity that is smaller than the escape velocity. If the collision occurs 

at an angle larger than the critical angle, the solitons simply go through each 

other unaffected - very similar to the behavior of Kerr solitons (the beams 

refract twice while going through each other’s induced waveguide but cannot 

couple light into it). If the collision occurs at ”shallow” angles, the beams 

can couple light into each other’s induced waveguide. Now if the waveguide 

can guide only a single-mode (a single bound state), the collision outcome 

will be elastic, essentially very similar to that of a similar collision in Kerr 

media (with the exception that now some very small fraction of the energy 

is lost to free radiation). However, if the waveguide can guide more than 

one mode, and if the collision is attractive, higher modes are excited in each 

waveguide and, in some cases, the waveguides merge and the solitons fuse to 

form one single soliton beam. Such a fusion process is always followed by a 

small energy loss to radiation waves, much like inelastic collisions between 

real particles. This naive picture of soliton interactions gives qualitative understanding 

of the complex behavior of soliton collisions, yet it is incomplete. 

In reality, the interacting solitons affect each other’s induced waveguide, and 

the true collision process is much more complicated. 

The first experimental papers on collisions between photorefractive solitons 

were also the first papers reporting fusion of solitons in any medium 

[49, 98] (in parallel to a similar observation with solitons in atomic vapor, 

for which the nonlinearity is also saturable). These experiments reported 

on incoherent collisions, between (2+1)D [49] and between (1+1)D solitons 

[98], during which at large collision angles the solitons passed through one 

another unaffected, whereas at shallow collision angles they fused to one another. 

Following these experiments, a team headed by W. Krolikowski studied 

coherent collisions and demonstrated fission of solitons (”birth” and annihilation 

of solitons [50, 51], which also constitutes the first observation of such 

effects in any solitonic system. Other groups have followed and mapped out 

coherent interactions between (1+1)D and (2+1)D solitons [99, 100]. All of 

these were collisions between solitons launched with trajectories in the same


plane. However, because the photorefractive nonlinearity is saturable, one 

can also look at collisions of (2+1)D solitons with trajectories that also do 

not lie in the same plane: full 3D interactions. When non-parallel solitons 

with trajectories that do not lie in the same plane are launched simultaneously, 

they interact (attract or repel each other) via the nonlinearity and 

their trajectories bend. The system possesses initial angular momentum: if 

the soliton attraction exactly balances the ”centrifugal force” due to rotation, 

the solitons can ”capture” each other into orbit and spiral about each 

other, much like two celestial objects or two moving charged particles. This 

idea was suggested in the context of coherent collisions. However, because coherent 

interactions are critically sensitive to phase, in this case solitons can 

never attain stable orbits, and spiral about each other. Instead, they always 

either fuse to form a single beam, or ”escape away” from each other. On the 

other hand, the purely attractive nature of the force between incoherent solitons 

and its independence of the relative phase of the two interacting solitons 

makes these ideal for the orbiting observation. Under proper initial conditions 

of separation and beam trajectories, solitons indeed capture each other into 

an elliptic orbit [48]. If the initial distance between the solitons is increased, 

their trajectories slightly bend toward each other, but their ”velocity” is 

larger than the escape velocity, and they do not form a ”bound pair”. On 

the other hand, if their separation is too small, they spiral on a ”converging 

orbit” and eventually fuse. In reality, the 3D spiraling-interaction mechanism 

is much richer and more complicated than initially thought. It turns out that 

the two spiraling-interacting solitons exchange energy by coupling light into 

each other’s induced waveguide. This is because the nonlinearity is saturable 

and the trajectories are at shallow angles. But, because the two interacting 

solitons have equal power, the energy exchange is symmetric. The energy exchanged 

is, of course, phase-coherent with its ”source” but phase incoherent 

with the soliton into which it was coupled. Thus, even though the solitons 

are initially incoherent with each other, the energy exchange induces partial 

coherence and thus contributes to the forces involved. The end result is that 

the two solitons orbit periodically about each other and at the same time 

exchange energy periodically, with the two periods (the spiraling period and 

the energy exchange period) being only indirectly related [101]. This complicated 

motion is stable over a wide range of parameters [101]. To some 

extent, whether or not the spiraling can go on indefinitely is yet an open 

question, because it is possible that, after long enough propagation distances 

(far beyond the present experimental reach), the solitons eventually merge 

[102, 103]. Another interesting feature of spiraling-interacting solitons is the 

fact that if one adds a tiny seed in one of the input solitons that is coherent 

with the other soliton, the relative phase between these coherent components 

(the seed and the other soliton) controls the outcome of the collision process, 

and can turn a spiraling motion into fusion or repulsion [101]. Altogether, 

the observation of spiraling-interacting solitons has introduced new concepts


to soliton physics: not only energy (power) and momentum are conserved, 

but also the conservation of angular momentum, which is the fundamental 

symmetry that enables spiraling. These pioneering experiments are a characteristic 

example of the contribution of photorefractive solitons to nonlinear 

science. 

Fig. 20. Collisions, from [49]. Top view of colliding incoherent diffracting beams 

and solitons for different relative angles. 

Fig. 21. Hybrid-soliton collisions, from [104]. (a) input launch of a two-dimensional 

and a one-dimensional diffracting beam, output intensity distribution with no applied 

field, output intensity distribution with applied self-trapping field; (b) same 

as (a) but for a smaller collision angle; (c-d) single beam self-trapping in the same 

experimental conditions. 

It is interesting to compare soliton and plane-wave interaction phenomenology 

in a photorefractive. The crossing of two plane waves, even at low 

angles of the order of fractions of a degree, leads to energy transfer, which, 

even if mediated by space-charge components of the order of η, brings to 

finite if not total coupling from one beam to the other. Two solitons, on the 

other hand, behave in a diametrically opposite manner: they cross each other 

without appreciable energy transfer, even when they are mutually coherent.


Fig. 22. Soliton fusion, from [50]. Two identical solitons fuse at the output of the 

sample when they are in-phase (a), whereas they repel when they are out-of-phase 

(b). 

Fig. 23. Soliton spiralling, from [48].(a) Beams A and B at the input face of the 

crystal, (b) the spiraling soliton pair after 6.5 mm of propagation, and (c) after 

13 mm of propagation. The centers of diffracting A and B are marked by white 

triangles. The white cross indicates the center of mass of the diffracting beams A 

and B in (b) and (c). 

The conundrum has a straightforward geometrical explanation: even though 

no soliton mechanism forbids the formation of a resonant coupling grating in 

the region where the two beams overlap, this overlap is spatially limited by 

the mutual angle θ in their propagation direction and their very narrow width 

∆x, which is typically no more than 20 wavelengths wide. Thus, in two-wavemixing 

terminology, even though the coupling coefficient γ can be reasonably 

high (e.g., at angles corresponding to the Debye length), the equivalent energy 

transfer γL, where L is the effective interaction distance in the sample, is 

negligible. Altogether, since self-lensing is a purely phase-modulating mechanism 

without energy coupling to new modes, soliton interactions, even in 

photorefractives (which could give rise to energy transfer driven by the diffusion 

field) is well described by means of soliton interaction forces based 

on wave-overlap integrals, and the intrinsic energy coupling mechanism that 

acts in two-wave-mixing is absent.


In the spirit of driving nonlinear soliton studies to their most extreme consequences, 

photorefraction forwards a unique possibility of studying hybridodimensional 

soliton collisions [104]. Namely, testing to what extent a soliton 

undergoes particle-like dynamics even during a collision between two solitons 

of different dimensionality. The possibility of carrying out these studies arises 

from the saturable nature of the nonlinearity, which facilitates stable (2+1)D 

solitons, as well as the quasi-stable propagation of (1+1)D solitons in 3D bulk 

media. Strictly speaking, the latter case is not stable in the absolute sense, 

and would, after a sufficiently large propagation distance, suffer from transverse 

instabilities and disintegrate into an array of 2D filaments. However, 

if the (1+1)D soliton is launched with parameters close enough to the soliton 

existence curve, and if its parameters are chosen such that the operation 

point is in the saturated regime, the ”instability length” is very long, and for 

all practical purposes such a (1+1)D soliton propagates in a stable fashion 

in the bulk. These conditions have indeed allowed to experimentally observe 

the collision between a ”needle soliton” and a ”stripe soliton” [104], resulting 

in a new avenue in the investigation of particle-like behavior of solitons at 

large. 

A more recent series of studies targeted collisions of solitons propagating 

in opposite directions [105, 106, 107, 108]. The concept actually applies to 

any soliton-supporting system and introduces a profoundly different scenario 

for interacting solitons [109]. Experimentally, thus far only incoherent collisions 

between such solitons have been studied [106, 108], whereas the more 

interesting case of coherent collisions [109, 107] is still unexplored [apart for 

the very specific case of a vector soliton formed by counter-propagating fields 

which has been recently demonstrated [110]; but this is not really a vector 

soliton, and not a soliton collision experiment]. In photorefractives, however, 

during an almost head-to-head collision, the interaction between the solitons 

occurs along a spatially extended region, hence spatially nonlocal effects, such 

as self-bending driven by diffusion fields, have an important impact on the 

collision dynamics. The experiments of [106, 108] show that the soliton collision 

dramatically modifies the self-bending of both solitons, in a way that 

can be directly utilized for all-optical beam deflection, steering and control. 

9 Vector and composite solitons 

In their most basic form, solitons are a coherent entity described by a single 

optical wavefunction. The simplest (so-called scalar) soliton occurs when 

the soliton constitutes of a single field, which populates the lowest mode of 

its own induced-waveguide. A more complex soliton, a vector soliton, occurs 

when more than one field populates the lowest mode. These were first suggested 

by Manakov in Kerr media, when self- and cross-phase modulation 

are equal. Vector solitons, however, can be also composite: they can be composed 

of optical fields that populate different modes of their jointly induced-


waveguide. In one realization, composite solitons are made of a bell-shaped 

(bright) component populating the lowest mode and a dark component being 

the second mode. A more interesting situation occurs when the field constituents 

of the composite-soliton populate different bound-modes of their 

jointly-induced potential. These composite, multi-mode, solitons can have 

two or more intensity humps, and can appear in (1+1)D and in (2+1)D, in a 

variety of intriguing combinations, including (2+1)D composite solitons carrying 

angular momentum within their field constituents. As will be explained 

below, almost all the experiments with vector solitons, and practically all the 

experiments with multi-mode solitons, were carried out in photorefractives. 

A key issue regarding a vector soliton is that interference terms between 

the soliton constituents should not contribute to the nonlinear index change 

(otherwise the induced potential would vary periodically during propagation). 

Thus, the field constituents making up a vector soliton could originate from 

orthogonal polarizations states, or from fields at widely-spaced frequencies. 

In the polarization-based technique, there are no interference terms, whereas 

in the widely spaced frequencies method, the interference terms are not synchronized 

with either of the soliton constituents. Photorefractives, however, 

have offered a much more appealing technique to generate vector solitons: 

making up a vector soliton from field constituents that are incoherent with 

one another, that is, their relative phases are randomly fluctuating. When the 

relative phase between the fields making up the soliton fluctuates much faster 

than the response time of the nonlinearity, their contribution to interference 

terms averages out to zero. This method, suggested by Christodoulides [111], 

has revolutionized the area of vector solitons. With this mutual incoherence 

method, first a degenerate (Manakov-like) soliton was observed [112], the 

same year that the first Manakov-type solitons were observed in Kerr media. 

This was followed by an observation of a vector soliton made up of a bright 

and a dark component [113]. Shortly thereafter, the first multi-mode/multihump 

solitons were observed [114]. In this multi-mode soliton experiment, 

the two input field distributions resembled the first and/or second and third 

modes of a slab waveguide. Interestingly, the experiment has also shown that 

it is possible to observe multi-mode solitons made up of solely higher modes 

(the second plus the third modes, trapped in their jointly-induced potential 

[114]). That is, the experiment has indicated that multi-mode solitons can 

exist and propagate in a stable fashion for distances much larger than the 

diffraction length, in spite of the fact they are made up of excited states only. 

Several years later, (2+1)D dipole-type composite solitons were also demonstrated 

experimentally [39, 41]. These vector solitons consist of a bell-shaped 

component jointly trapped with a 2D dipole mode. The ability to generate 

(2+1)D composite solitons opens up a whole new range of possibilities that 

has no counterpart in a lower dimensionality. One fascinating example is the 

recently observed rotating ”propeller” soliton [40]. This is a composite soliton 

made of a rotating dipole component jointly trapped with a bell-shaped


component. It carries and conserves angular momentum, although its constituents 

exchange angular momentum as they propagate. The soliton turns 

out to be generically robust, so much that even during inelastic collisions 

with other composite solitons, the collision products are predicted to be also 

composite rotating propeller solitons [115]. 

The relatively ease with which vector and composite solitons are generated 

in photorefractives, by employing the mutual incoherence technique, 

has also led to a series of experimental efforts demonstrating interactioncollisions 

between vector solitons. It turns out that temporal optical vector 

solitons, and non-optical vector solitons are very difficult to generate, hence 

the collision experiments with optical spatial vector soliton were truly pioneering. 

For example, it has been predicted, more than two decades ago, that 

collisions between Manakov-like vector solitons give rise to a symmetric exchange 

of energy between the soliton constituents. This elegant phenomenon 

was observed only recently, with photorefractive vector (Manakov-like) solitons 

[116, 117]. The energy-exchanges between the soliton components (which 

have nothing to do with photorefractive two-wave-mixing) have direct implications 

in a new form of reversible computing, in which a ”state” is coded 

as the ratio between the soliton components [118]. In this scheme, computation 

is performed through the energy-exchange interactions (in which the 

”states” change) during collisions between vector solitons. The experiments 

have shown that indeed, not only such symmetric energy exchanges do occur, 

but also information can be transferred through a series of cascaded collisions 

between vector solitons [116, 117]. 

Finally, photorefractives were also the means for experimental studies 

of interaction-collisions between multi-mode solitons, in which shape transformations 

were observed [119]. These were the first ever experiments with 

collisions of multi-mode solitons. 

The general ideas behind multi-component vector solitons proved invaluable 

for later developments and in particular to the area of incoherent solitons 

discussed in the next section. 

10 Incoherent solitons: self-trapping of 

weakly-correlated wavepackets 

Until 1996, the commonly held belief was that all soliton structures should be 

inherently coherent entities. In that year however, an experiment carried out 

at Princeton demonstrated beyond doubt that self-trapping of a partially 

spatially-incoherent light beam [42] is in fact possible, if the nonlinearity 

has a non-instantaneous temporal response. In that experiment, the optical 

beam was quasi-monochromatic, but partially spatially-incoherent and the 

nonlinear medium was photorefractive, with a response much slower than 

the characteristic time of the phase fluctuations in the incoherent beam. The 

resultant self-trapped beam is now commonly referred to as an ”incoherent


Fig. 24. The observation of a multimode soliton, from [114],with mode 2/mode 1 = 

1.0 (a),(b) input (18 µm FWHM) and linear diffraction (27 µm) of mode 1; (c),(d) 

input (28 µm) and linear diffraction (62 µm) of mode 2; (e),(f) combined input 

(26 µm) and combined linear diffraction (58 µm); (g) composite soliton formed (26 

µm) with application of 800 V/4 mm; (h) first mode obtained after quickly blocking 

second mode; (i) steady state of first mode alone with nonlinearity on; (j) second 

mode obtained after quickly blocking first mode; and (k) steady state of second 

mode alone with nonlinearity on. 

soliton” or a ”random-phase soliton”. One year later, the same group observed 

a similar self-trapping effect with white light from an incandescent light bulb 

emitting light with a wide 380-720 nm spectrum of wavelengths. This was the 

first observation of a self-trapped beam made of light both temporally and 

spatially incoherent: a white light soliton [43]. These two experiments have 

set the basis for a new field of study in nonlinear optics: incoherent solitons. 

In yet another experiment, self-trapping of dark incoherent ”beams”, i.e., 1D 

or 2D ”voids” nested in a spatially incoherent beam, was also demonstrated 

[47]. 

For self-trapping of an incoherent beam (an incoherent soliton) to occur, 

several conditions must be satisfied. First, the nonlinearity must be noninstantaneous 

with a response time that is much longer than the phase fluctuation 

time across the optical beam. Such a nonlinearity responds to the 

time-averaged envelope and not to the instantaneous ”speckles” that constitute 

the incoherent field. Second, the multi-mode (speckled) beam should 

be able to induce, via the nonlinearity, a multi-mode waveguide. Otherwise,


if the induced waveguide is able to support only a single guided mode, the 

incoherent beam will simply undergo spatial filtering, thus radiating all of its 

power but the small fraction that coincides with that guided mode. Third, as 

with all solitons, self-trapping requires self-consistency: the multi-mode beam 

must be able to guide itself in its own induced waveguide (pages 86-125 in 

Ref.[4]). 

The understanding of how such an incoherent soliton can form also raises 

some intriguing aspects in comparison to other nonlinear phenomena. Conventional 

nonlinear optical effects are a consequence of strong material response 

to (intense) optical excitation, but their macroscopic manifestation is 

generally the result of distributed enhancement effects, in which weak scattering 

is amplified by the cooperative excitation of large portions of material. 

This is the basis, for example, for efficient harmonic generation and wavemixing. 

The distributed effect is induced by an extended spatial and temporal 

coherence, itself transferred by the coherence of the coupling waves. 

The discovery of incoherent solitons has made evident the basic fact that 

solitons naturally break this scheme. Even though they do result from nonlinearity, 

their nature has nothing to do with the constructive interference of 

distributed effects. On the contrary, it is an intrinsically local effect where 

no coherence transfer mechanism intervenes. Thus, for the more diverse nonoptical 

media that support solitons, the very concept of wave is an abstract 

average envelope of intrinsically uncorrelated underlying motion, that attains 

physical meaning when the wave-medium interaction does not react to the 

erratic fluctuations of the wave constituent. 

The experiments demonstrating incoherent solitons have taken the solitons 

community by surprise, because typically, in most of soliton research 

(also beyond Optics), all experiments were preceded by a theory predicting 

the main effects. The experiments demonstrated beyond doubt that incoherent 

solitons indeed exist. Yet, at the time, something quite important was 

still missing: a theory! Unlike the case of coherent solitons, where the evolution 

equation can be straightforwardly derived by adding the nonlinearity 

to the paraxial equation of diffraction, the description of incoherent solitons 

was far from being clear. The experiments were of course based on insight 

and intuition, but then again they gave only few clues, if any, as to how one 

could develop a theory. Only one thing was certain - the theory of incoherent 

solitons had to be derived from first principles. Within a year, two different 

theories were developed to describe incoherent solitons: the coherent density 

theory [120] and the modal theory [121]. The coherent density theory is, by its 

very nature, a dynamic approach that is better suited to study the evolution 

dynamics of incoherent solitons, their interactions, instabilities etc, as they 

occur in experimental set-ups. In this formalism, the incoherent field is described 

by means of an auxiliary non-observable function from where one can 

deduce the optical intensity as well as the associated correlation statistics. 

The modal theory, on the other hand, by virtue of its inherent simplicity be-


came the method of choice in terms of identifying incoherent solitons, their 

range of existence and correlation properties. One year later, yet another 

theory was proposed: the theory describing the propagation of the mutual 

coherence [122]. Interestingly enough, even though at first sight these three 

theoretical approaches seem to be dissimilar, they are in fact formally equivalent 

to one another [123]. All describe quasi-monochromatic yet partially 

spatially incoherent solitons: they explain the behavior of such entities, provide 

their statistical properties, and predict their behavior as they interact 

with one another. As such, they became a very powerful analytical and numerical 

tool. More recently, these theories were expanded to describe white 

light solitons: solitons that are made of temporally and spatially incoherent 

light [124, 125]. 

The pioneering experiments of Refs.[42, 43, 47] that were the first to 

show the unexpected fact that random-phase solitons can exist with both 

spatial and temporal incoherence, opened the way for several other important 

results. These include for example, anti-dark incoherent states [126], 

elliptic incoherent solitons [127, 128], coherence control using interactions 

of incoherent solitons [129, 130], and more. Especially noteworthy are the 

fundamental studies on modulation instability of incoherent waves and spontaneous 

pattern formation in weakly correlated system. It has been found, 

theoretically and experimentally, that such weakly correlated systems indeed 

exhibit modulation instability, and a uniform (homogeneous) partially incoherent 

wavefront breaks up into an array of soliton-like filaments. However, 

for this to occur the nonlinearity needs to exceed a threshold value determined 

by the coherence properties of the waves [131, 132, 133, 134, 135, 136]. This 

fact stands in sharp contradistinction with coherent systems, for which there 

is no such threshold for modulation instability (in a coherent self-focusing 

system, a uniform wave always breaks up into an array of soliton-like structures). 

Once this threshold is exceeded, the partially incoherent uniform wave 

breaks up into an array of localized structures, each behaving as an incoherent 

soliton. These solitonic ”breakup products” interact with one another 

in an incoherent fashion, exerting long-range attraction on each other. After 

sufficiently large propagation distance, they aggregate and form clusters 

of fine-scale structures, leaving large voids between adjacent soliton clusters 

(aggregates of solitons) [137, 138]. More recent work along these veins is the 

prediction of modulation instability of white light [139], along with its very 

recent experimental observation [140], and the work on incoherent pattern 

formation in cavities [141, 142, 143, 144]. 

Studies on incoherent solitons and incoherent pattern formation have established 

that these are not some kind of esoteric creatures, specific to photorefractives, 

but are in fact a general and rich new class of solitons, whose 

existence is relevant to many other diverse fields beyond nonlinear optics. For 

example, incoherent modulation instability effects, soliton clustering and incoherent 

pattern formation, relate to many systems in nature: from clustering


in a cooled atomic gas to self-supported ”stripes” of electrons in semiconductors, 

as well as to gravitational-like effects. In fact, the underlying physics 

relates to any weakly-correlated wave-system having a non-instantaneous 

nonlinearity. Altogether, it is fair to say that incoherent solitons are most 

probably the single most important discovery made with self-trapping effect 

in photorefractive systems. It has introduced a new concept into soliton research, 

and has many implications beyond optics, into other arenas where 

random phase waves and nonlinearities coexist. 

11 Applications 

11.1 Passive devices 

The most basic functionality afforded by a soliton in any physical system, 

is the transfer of a localized energy/information bearing wave perturbation 

along an otherwise dispersive propagation. For an optical spatial soliton, 

this translates into the fact that far-field effects are absent, and the spatial 

resolution, its phase curvature and coherence properties, and the transverse 

distribution of energy, remain unaltered: a guided propagation in an otherwise 

bulk and homogeneous medium, the guide itself being induced by the modes 

it supports. 

For photorefractive solitons, the composite nonlinearity that intervenes 

allows for a series of useful attributes. For one, a photorefractive soliton 

can passively guide a second beam of longer wavelength, on consequence of 

the fact that whereas self-trapping is a product of nonlinearity, the indexmodulating 

system is only weakly wavelength dependent, and will act on an 

infrared beam in much the same manner that it acts on the soliton [145, 146, 

147, 148, 149, 150, 151]. Not being of nonlinear origin, this phenomenon does 

not depend on the intensity of the guided signal, and, in the measure in which 

the deposited charge displacement which accompanies the soliton does not 

redistributed, does not even require the presence of the visible nonlinear wave. 

This property can be used to integrate a material into a fiber or waveguide 

device without the development of a crystal-specific technique to grow or 

tailor a waveguide, the sample itself serving either as an electro-optic phasetransducer, 

the waveguide not being greatly modified by the application of 

an arbitrary bias for linear electro-optic response, or simply as redirecting 

component. 

Passive waveguiding gives us the opportunity of differentiating between a 

nonlinear and a linear behavior. Consider, for example, a dark soliton, which 

consists of a nondiffracting intensity notch engendered by a π phase jump. 

In the very same conditions that lead to its formation, it can passively guide 

a bright longer wavelength mode, even though such a photoactive mode (at 

the shorter wavelength) could never self-trap in the same conditions.


Evidently, more complicated passive devices can be demonstrated, such 

as reconfigurable directional couplers based on two bright solitons formed 

in close proximity, Y-junctions, along with more complex multisoliton structures 

and hybrid soliton-fabricated-waveguides systems [152, 153, 154, 155, 

156, 157, 158]. Moreover, in conditions in which charge redistribution cannot 

be avoided, a considerable advantage can be obtained by implementing the 

ferroelectric fixing of the routing device [159]. 

11.2 Active devices 

A second functionality is based on the fact that photorefractives offer alloptical 

functionality even at low intensities, even though this is burdened by 

slow time response [160, 161, 162, 163, 164, 165]. For example, logic operations 

can be carried out simply by having two solitons interact, or by modulating 

two different components of a single vector soliton. We might note that a 

truly all-optical soliton device must imply a partial absorbtion, and this does 

not generally spouse implementation. A slightly less ambitious alternative is 

to use the all-optical operation to steer non photoactive signals. 

A second form of active device is that for which soliton dynamics are 

controlled externally by means of a modulation in the supporting physical 

parameters. Thus for example, the output direction of propagation can be 

changed by changing the bias voltage, as a consequence of self-bending [21]. 

Once again, whereas the signal can be delivered in a fast capacity-limited 

regime, the optical response will be dominated by the photorefractive time 

constants. 

11.3 Electro-optic manipulation 

Passive electro-optic effects have been investigated for soliton-deposited 

space-charge in paraelectric samples [166]. This allows the fast capacitylimited 

manipulation of optical circuitry through the sole electro-optic effect, 

much in the same manner as electro-holographic technology. 

In order to grasp the phenomenon, note that in a linear electro-optic 

response, once a soliton has been formed through a self-trapping ∆ns, the 

application of an arbitrary control external bias in combination with E0 leads 

to a ∆n(Econ) ∝ (E + Econ), which simply changes the constant pedestal on 

which the soliton guiding pattern induced by E is embedded. This means, for 

example, that passive guiding can be achieved also for a zero applied external 

field. For a paraelectric, which is characterized by a quadratic response, 

∆n(Econ) ∝ (E + Econ) 2 = E 2 +2EEcon + E 2 con. The second mixed term allows 

for an electro-optic distortion of the pattern, which, not implying charge 

redistribution, is a purely spatially resolved electro-response. This has allowed 

the demonstration of a series of beam manipulation devices, culminating in 

a two-needle switching device [167] (see Fig.25).


11.4 Nonlinear frequency conversion in waveguides induced by 

photorefractive solitons 

The most promising application of waveguides induced by photorefractive 

solitons is nonlinear frequency conversion. The conversion efficiency in χ 2 

processes is proportional to the intensity of the pump beam, so it is desirable 

to work with very narrow beams. One easy way to achieve that is to 

use a focused pump beam. However, in a bulk crystal, the more focused a 

beam is, the faster it diffracts, and diffraction limits the frequency conversion 

efficiency because as the interacting beams diffract, (1) their intensities 

decrease, and (2) the phase-matching condition cannot be satisfied across 

their entire cross section. Therefore, using waveguides for frequency conversion 

can greatly improve the conversion efficiency. But thus far it has been 

difficult to fabricate waveguides from most materials that allow for phase 

matching, and two-dimensional waveguides are especially difficult to make. 

Now, (2+1)D photorefractive solitons induce 2D waveguides, and almost all 

photorefractives are highly efficient in χ 2 frequency conversion. In waveguides, 

phase-matching should take place among the propagation constants 

of the guided modes, and is typically obtained through birefringence or periodic 

poling. In a fabricated waveguide, however, the structure is fixed, so 

tuning techniques rely on varying the temperature, or on lateral translation 

in structures with several periods of poling parallel to one other. But waveguides 

induced by photorefractive solitons offer much flexibility because their 

waveguide structure and propagation axis (with respect to the crystalline 

axes) can be modified at will and in real time. Working with photorefractive 

solitons, one can achieve wavelength tunability while avoiding diffraction by 

simply rotating the crystal and launching a soliton in the new direction. One 

can also fine-tune the frequency conversion process by changing the propagation 

constants of the guided modes through varying the intensity ratio and 

external voltage, allowing tuning with no mechanical movements. 

The first step in the direction of nonlinear frequency conversion in waveguides 

induced by photorefractive solitons was the demonstration of efficient 

second-harmonic generation [168, 169, 170]. The experiment have shown that 

the conversion efficiency can be considerably increased [168], and high tunability 

can be obtained by rotating the crystal [169]. However, a much more 

important scenario occurs in a soliton-based optical parametric oscillator 

(OPO). In an OPO, the threshold pump power is dependent on the signal 

gain per pass through the crystal. To lower the threshold, one has to increase 

the signal gain per pass. A waveguide that confines the pump beam 

as well as the signal and idler in a small area is one very effective way to 

achieve this. Consider a Gaussian beam at the pump frequency launched 

into a nonlinear crystal and assume that phase matching is satisfied at the 

waist, located at the input surface. The threshold pump power is proportional 

to [z0 arctan 2 (L/z0)+ln 2 (1 + L 2 /z 2 0)/4] −1 , where z0 is the Rayleigh 

(diffraction) length of the beam and L is the crystal length. For a given L,


there exists an optimum beam size for minimum threshold pump power, when 

z0 = L/2.84. However, if a waveguide is used to keep all beams at the same 

widths throughout propagation in the crystal, the threshold is simply proportional 

to z0/L 2 , which continues to decrease as we focus the beam more. 

The minimum threshold is determined by the smallest size of the waveguide 

that can be made. To estimate the improvement, consider a focused Gaussian 

beam with a minimum beam waist of 21 mm and a 15 mm long crystal. An 

OPO constructed in a waveguide has a threshold pump power 60 percent 

lower than that of an OPO with the same beam waist in bulk. Therefore, 

in a waveguide OPO the signal gain per pass can be considerably improved, 

and the threshold pump power can be substantially lowered for the same cavity 

loss. This generic idea was recently demonstrated with a doubly-resonant 

optical parametric oscillator in a waveguide induced by a photorefractive soliton 

[171]. The OPO threshold was considerably lowered by constructing it 

within a soliton-induced waveguide. This technique should work even better 

with singly resonant OPOs and it can substantially reduce the threshold 

pump power when very narrow solitons are employed. For example, using a 

soliton beam with a beam waist of 8 mm and a 15 mm long crystal, the OPO 

threshold pump could be reduced to only 3.5 percent of that for an OPO in 

the same nonlinear medium, using the same mirrors but without the soliton. 

Fig. 25. Electro-optic switching, from [167] 

12 New frontiers in photorefractive solitons and 

concluding remarks 

As we hope transpires from the brief review, there is still much to be understood 

as to the mechanisms underlying spatial photorefractive self-trapping, 

from the formation of 2+1D solitons to quasi-steady-state solitons, from dark 

incoherent solitons to spontaneous self-trapping. The field, however, is in 

constant expansion, and we cannot refrain from mentioning the important


Fig. 26. Second harmonic generation in photorefractive solitons, from [168]. 

Fig. 27. Second harmonic conversion efficiency increase mediated by spatial solitons, 

from [168] 

achievements in the field of soliton lattices and discrete soliton observation 

obtained in photorefractive SBN [172, 173, 174, 175, 176, 177], and in the 

trapping of more exotic excitations such as rings [178]. 

Although we have attempted to give a detailed account of all aspects 

of photorefractive self-trapping, the field has undergone such a rapid and 

extensive evolution that we can hardly guarantee that all contributions have 

been cited and described. Perhaps this is yet another accomplishment of 

photorefraction, that is, having given to the soliton science community a 

powerful and accessible tool with which to further its research and develop 

new and possibly useful scientific and practical tools. 

13 Dedication 

Moti Segev would like to dedicate this Chapter to his friend Galit Staier, who 

was critically wounded in the murderous Palestinian suicide attack on October 

4, 2003, in the Maxim restaurant in Haifa, Israel. In that suicide bomb a 

Palestinian woman blew herself up killing 21 civilians. Galit has has lost her 

11-year old son Assaf Staier, her father Ze’ev Almog (71), her mother Ruth


Almog (70), her brother Moshe (43), and his son Tomer Almog (9). Moshe’s 

other son, Oran Almog (11), was critically wounded, while Moshe’s wife, Orly 

Almog, and their daughter Adi Almog (4), were moderately wounded as well. 

Galit and the survivors of the Staier-Almog family are now slowly recovering. 

We all hope that Galit will fully recover soon, for her husband (Moti’s friend) 

Ofer Staier, for their son Omri, and for her family and friends. 

The Authors join in expressing the hope that the sacrifice of the Staier- 

Almog family lighten the way to a real peace, free of horrible actions against 

humanity, such as suicide bombings. 

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